Subgraphs and Colourability of Locatable Graphs

We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a different direction of topic, we also start research in the direction of colourability of locatable graphs, we also show that every locatable graph is 4-colourable, but not necessarily 3-colourable.Comment: 25 page

Thresholds and the structure of sparse random graphs

In this thesis, we obtain approximations to the non-3-colourability threshold of sparse random graphs and we investigate the structure of random graphs near the region where the transition from 3-colourability to non-3-colourability seems to occur. It has been observed that, as for many other properties, the property of non-3-colourability of graphs exhibits a sharp threshold behaviour. It is conjectured that there exists a critical average degree such that when the average degree of a random graph is around this value the probability of the random graph being non-3-colourable changes rapidly from near 0 to near 1. The difficulty in calculating the critical value arises because the number of proper 3-colourings of a random graph is not concentrated: there is a `jackpot' effect. In order to reduce this effect, we focus on a sub-family of proper 3-colourings, which are called rigid 3-colourings. We give precise estimates for their expected number and we deduce that when the average degree of a random graph is bigger than 5, then the graph is asymptotically almost surely not 3-colourable. After that, we investigate the non-$k$-colourability of random regular graphs for any $k \geq 3$. Using a first moment argument, for each $k \geq 3$ we provide a bound so that whenever the degree of the random regular graph is bigger than this, then the random regular graph is asymptotically almost surely not $k$-colourable. Moreover, in a (failed!) attempt to show that almost all 5-regular graphs are not 3-colourable, we analyse the expected number of rigid 3-colourings of a random 5-regular graph. Motivated by the fact that the transition from 3-colourability to non-3-colourability occurs inside the subgraph of the random graph that is called the 3-core, we investigate the structure of this subgraph after its appearance. Indeed, we do this for the $k$-core, for any $k \geq 2$; and by extending existing techniques we obtain the asymptotic behaviour of the proportion of vertices of each fixed degree. Finally, we apply these results in order to obtain a more clear view of the structure of the 2-core (or simply the core) of a random graph after the emergence of its giant component. We determine the asymptotic distributions of the numbers of isolated cycles in the core as well as of those cycles that are not isolated there having any fixed length. Then we focus on its giant component, and in particular we give the asymptotic distributions of the numbers of 2-vertex and 2-edge-connected components

Waiter–Client and Client–Waiter games

In this thesis, we consider two types of positional games; $Waiter$-$Client$ and $Client$-$Waiter$ games. Each round in a biased ($a$:$b$) game begins with Waiter offering a+b free elements of the board to Client. Client claims $a$ elements among these and the remaining $b$ elements are claimed by Waiter. Waiter wins in a Waiter-Client game if he can force Client to fully claim a $winning$ $set$, otherwise Client wins. In a Client-Waiter game, Client wins if he can claim a winning set himself, else Waiter wins. We estimate the $threshold$ $bias$ of four different ($1$:$q$) Waiter-Client and Client-Waiter games. This is the unique value of Waiter's bias $q$ at which the player with a winning strategy changes. We find its asymptotic value for both versions of the complete-minor and non-planarity games and give bounds for both versions of the non-$r$-colourability and $k$-SAT games. Our results show that these games exhibit a heuristic called the $probabilistic$ $intuition$. We also find sharp probability thresholds for the appearance of a graph in the random graph $G$($n$,$p$) on which Waiter and Client win the ($1$:$q$) Waiter-Client and Client-Waiter Hamiltonicity games respectively

On the chromatic number of a random hypergraph

We consider the problem of $k$-colouring a random $r$-uniform hypergraph with $n$ vertices and $cn$ edges, where $k$, $r$, $c$ remain constant as $n$ tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case $r=2$, must have one of two easily computable values as $n$ tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs.Comment: 45 pages, 2 figures, revised versio